Using database.read() will read your text file and return a single string containing the entire file. Looping over this string will return each character within that string individually, so you won't find 'MY STRING' in it. Simply look for 'MY STRING' within the file as a whole like so:

with open("FILE PATH", "a+") as database:
if 'MY STRING' not in database.read():
database.write('MY STRING')

You don't need the first if statement seeing as it's only a continue, like this you only have one if, making it a little more efficient.